Math 39032 Mathematical Modelling Of Finance Take-Home Cours

Math39032 Mathematical Modelling Of Financetake Home Courseworkthis As

This assignment involves modeling financial derivatives, specifically options involving two underlying assets S1 and S2, their volatilities, dividends, and related processes. The questions cover the justification of stochastic processes used, hedging strategies, derivation of the governing PDE, analysis of an exchange option, and deriving explicit solution formulas.

Answer all questions provided, clearly explaining your working, assumptions, and referencing textbooks or journal articles where appropriate. The solution must be concise, well-structured, and suitable for submission as a PDF file less than 6 pages.

Paper For Above instruction

Introduction

In quantitative finance, modeling the evolution of asset prices is fundamental for derivative pricing and risk management. The commonly used framework involves stochastic processes, especially geometric Brownian motion, to model asset dynamics with volatility and dividend considerations. This paper addresses the modeling of a two-asset option, including justification of stochastic processes, hedging strategies, derivation of the governing PDE, analysis of exchange options, and derivation of explicit valuation formulas.

Stochastic Modeling of Underlyings

The process for the underlying asset Si is given by the stochastic differential equation (SDE)

dS_i = (μ_i - D_i)S_i dt + σ_i S_i dW_i

where the Wiener processes dWi are independent, satisfying E[dWi²] = dt and E[dW1dW2] = 0. The use of geometric Brownian motion arises from its properties of modeling continuous, multiplicative price changes that align with observed asset returns, which are approximately log-normal and exhibit stochastic volatility. The drift term (μi-Di) reflects the expected growth rate reduced by dividends, which effectively lower the asset's price appreciation rate, consistent with the no-arbitrage principle and market efficiency assumptions (Shreve, 2004; Baxter & Rennie, 1996).

Hedging Portfolio Construction

Consider a portfolio \(\Pi = V - \Delta_1 S_1 - \Delta_2 S_2\). To eliminate risk from stochastic fluctuations, the portfolio should be locally riskless and hedged against asset price changes. Applying Itô's Lemma to V(S1, S2, t), the differential is:

dV = ∂V/∂t dt + ∂V/∂S_1 dS_1 + ∂V/∂S_2 dS_2 + ½ ∂²V/∂S_1² (dS_1)² + ½ ∂²V/∂S_2² (dS_2)² + ∂²V/∂S_1 ∂S_2 dS_1 dS_2

Hedging eliminates the stochastic components, leading to choosing \(\Delta_1\) and \(\Delta_2\) such that the stochastic terms cancel:

Δ_1 = ∂V/∂S_1, \quad Δ_2 = ∂V/∂S_2

Thus, the portfolio becomes perfectly hedged when the holdings in the assets equal these partial derivatives, removing stochastic risk (Hull, 2018).

Derivation of the Governing PDE

By applying Itô's lemma, the riskless portfolio's expected return—compared with the risk-free rate—imposes the PDE (via the no-arbitrage condition):

∂V/∂t + ½ σ_1² S_1² ∂²V/∂S_1² + ½ σ_2² S_2² ∂²V/∂S_2² + (r - D_1) S_1 ∂V/∂S_1 + (r - D_2) S_2 ∂V/∂S_2 - rV = 0

This PDE governs the evolution of the option value V(S1, S2, t) within the market assumptions (Musiela & Rutkowski, 2005).

Analysis of the Exchange Option

Considering the payoff for the exchange option at expiration T:

V(S_1, S_2, T) = max(S_1 - S_2, 0)

and the boundary conditions: V → 0 as S1 → 0, V → S1 e−D1(T−t) as S2 → 0, and V → S1 e−D1(T−t) as S1 → ∞, the PDE reduces when assuming a solution of the form V = S2 H(ξ, t), with ξ = S1/S2.

Substituting this form into the PDE and identifying the parameters leads to a simplified PDE for H(ξ, t):

∂H/∂t + ½ σ̂² ξ² ∂²H/∂ξ² + (D_2 - D_1) ξ ∂H/∂ξ - D_2 H = 0

where \(\sigmâ\) is the combined volatility derived from \(\sigma_1\) and \(\sigma_2\); specifically, \(\sigmâ^2 = \sigma_1^2 + \sigma_2^2\) if assets are uncorrelated.

Boundary Conditions for H

  • H(ξ, T) = max(ξ − 1, 0)
  • As ξ → 0, H(ξ, t) → 0
  • As ξ → ∞, H(ξ, t) → ξ

Explicit Solution of the Exchange Option

Following Margrabe (1978), the solution for V in terms of the standard normal cumulative distribution function N(d) is:

V(S_1, S_2, t) = S_1 e^{−D_1(T−t)} N(d_1) - S_2 e^{−D_2(T−t)} N(d_2)

where

d̂_1 = \frac{\ln(S_1 / S_2) + (D_2 - D_1) (T−t) + ½ σ̂^2 (T−t)}{σ̂ √(T−t)}, \quad d̂_2 = d̂_1 - σ̂ √(T−t)

and \(\sigmâ^2\) is the combined volatility, possibly incorporating correlation if present. These formulas provide a closed-form valuation for the exchange option in a complete market setting (Boyle, 1988; Hull, 2018).

Conclusion

This paper has outlined the theoretical framework for modeling a two-asset option, detailed the derivation of the relevant PDEs, and demonstrated an explicit solution for exchange options. These tools are fundamental for effective hedging and valuation strategies in multi-asset financial markets, highlighting the importance of stochastic calculus and boundary conditions in derivative pricing.

References

  • Baxter, M., & Rennie, A. (1996). Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press.
  • Boyle, P. (1988). An Examination of the Analytic Approximations to the Price of American Options. Journal of Financial Economics, 22(2), 165-189.
  • Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
  • Margrabe, W. (1978). The Value of Exchange Options. Journal of Finance, 33(1), 177-186.
  • Musiela, M., & Rutkowski, M. (2005). Martingale Methods in Financial Modelling. Springer.
  • Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer.

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